We consider multidimensional discrete valued random walks with nonzero drift killed when leaving general cones of the euclidian space. We find the asymptotics for the exit time from the cone and study weak convergence of the process conditioned on not leaving the cone. We get quasistationarity of its limiting distribution. Finally we construct a version of the random walk conditioned to never leave the cone.
We prove invariance principles for a multidimensional random walk conditioned to stay in a cone. Our first result concerns convergence towards the Brownian meander in the cone. Furthermore, we prove functional convergence of h-transformed random walk to the corresponding h-transform of the Brownian motion. Finally, we prove an invariance principle for bridges of a random walk in a cone.1991 Mathematics Subject Classification. Primary 60G50; Secondary 60G40, 60F17.
We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.Résumé. -Nous déterminons le comportement asymptotique de la fonction de Green pour des marches aléatoires à dérive nulle confinées dans des cônes convexes multidimensionnels. Comme conséquence, nous prouvons qu'il existe une unique fonction harmonique discrète et positive pour ces processus aléatoires, c'est-à-dire que leur frontière de Martin se réduit à un singleton.
We prove the existence of uncountably many positive harmonic functions for random walks on the euclidean lattice with non-zero drift, killed when leaving two dimensional convex cones with vertex in 0. Our proof is an adaption of the proof for the positive quadrant from [Ignatiouk-Robert, Loree]. We also make the natural conjecture about the Martin boundary for general convex cones in two dimensions. This is still an open problem and here we only indicate where the proof technique for the positive quadrant breaks down.
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